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Biometrika 1997 84(2):249-268; doi:10.1093/biomet/84.2.249
© 1997 by Biometrika Trust
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Markov chain models for threshold exceedances

RICHARD L. SMITH, JONATHAN A. TAWN and STUART G. COLES

Department of Statistics, University of North Carolina Chapel Hill, North Carolina 27599-3260, U.S.A. e-mail: rs{at}stat.unc.edu
Department of Mathematics and Statistics, Luncaster University Lancaster, LA1 4 YF, U.K. e-mail: j.tawn{at}lancaster.ac.uk

In recent research on extreme value statistics, there has been an extensive development of threshold methods, first in the univariate case and subsequently in the multivariate case as well. In this paper, an alternative methodology for extreme values of univariate time series is developed, by assuming that the time series is Markovian and using bivariate extreme value theory to suggest appropriate models for the transition distributions. A new likelihood representation for threshold methods is presented which we apply to a Markovian time series. An important motivation for developing this kind of theory is the possibility of calculating probability distributions for functionals of extreme events. We address this issue by showing how a theory of compound Poisson limits for additive functionals can be combined with simulation to obtain numerical solutions for problems of practical interest. The methods are illustrated by application to temperature data.

Key Words: Extreme value theory • Generalised Pareto distribution • Markov chain • Threshold model


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